# Questions tagged [triangulated-categories]

A triangulated category is an additive category equipped with the additional structure of an autoequivalence (called the translation functor) and a class of of triangles satisfying certain axioms.

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### Compatibility of different exchange structures $\operatorname{Ex}^*_{\#},\operatorname{Ex}^*_*,\operatorname{Ex}_{\# *}$

Let $\mathcal{Cat}$ denotes the $2$-category of small categories and $\mathscr{S}=\mathrm{Sch}/S$ be some category of schemes over a given scheme $S$, consider a $2$-functor $\mathscr{M}:\mathscr{S}^{...

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### structure in triangulated category similar to t-structure

It’s well known that the heart of a t-structure is an abelian category. My question is that can we find some structure on a triangulated category which can “produce” an exact category in analogy with ...

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### For reduced Noetherian ring $R$, the bounded derived category of $\operatorname{mod}R$ having strong generator implies $R$ has finite Krull dimension?

$\DeclareMathOperator\mod{mod}$Let $R$ be a reduced commutative Noetherian ring, let $\mod R$ be the abelian category of all finitely generated $R$-modules and let $D^b(\mod R)$ be its bounded derived ...

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### Mutations in triangulated category and cluster algebra

Let $\mathcal{D}$ be an enhanced triangulated category (basically meaning that $\operatorname{Hom}$'s are complexes). There is the notion of mutation in an enhanced triangulated category: given a full ...

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### Which "tensor" endofunctors on triangulated categories are essentially exact?

Assume that $T$ is a symmetric monoidal triangulated category, and $X$ is an object in it. Then the functors $X\otimes -$ and $-\otimes X: T\to T$ are not necessarily exact since they send ...

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### Slope-stability, tilt-stability, and Bridgeland stability

Following the definition of slope-stability ($\mu$-stability) and tilt-stability ($\nu$-stability) on page 8 of https://arxiv.org/abs/1410.1585, does an object's tilt-stability imply its slope-...

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### How to compute the higher $K$-theory of a triangulated category having a semi-orthogonal decomposition?

I am starting to learn the $K$-theory of triangulated categories and is stuck with the following.
Let $\mathcal{T}$ be a triangulated category having a semi-orthogonal decomposition $\langle \mathcal{...

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### Direct images commute with homotopy colimits

In Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (II), Ayoub defined the notion of a stable homotopical algebraic derivators; roughly, for a ...

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### "Essential injectivity" of Balmer spectra

Let $(\mathcal T, \otimes)$ be a tensor tringulated (tt-)category. Balmer defined a functor from the category of tt-categories to the category of locally ringed spaces, called the Balmer spectra or tt-...

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### Triangulated structure on complexes of mixed Hodge structures

I'm trying to read parts of the Peters Tata Lectures on "Motivic Aspects of Mixed Hodge structures"
One aspect I don't really understand is the construction of the ''mixed cone'' for ...

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### Moral reason for negative sign in rotation axiom for triangulated categories

I would like to know if there is a "moral" reason why in the definition of triangulated categories the "rotation axiom" TR2 requires that we have to add a negative sign to an arrow ...

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### Can higher G-theory of Noetherian schemes be computed by derived categories?

Recently I learned from the Stacks project that for every abelian category ${\mathcal A}$, there is a natural isomorphism $K_0({\mathcal A})\cong K_0(D^{b}(\mathcal A))$.
When we set $\mathcal A$ to ...

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### When is an $\infty$-categorical localization of an additive 1-category enriched in topological abelian groups?

Let $\mathcal A$ be an additive 1-category, equipped with some class of weak equivalences $\mathcal W$. Let $\mathcal A[\mathcal W^{-1}]$ be the localization of $\mathcal A$ at $\mathcal W$ (so $\...

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### Why the stable module category?

Let $R$ be a ring (usually assumed to be Frobenius). The stable module category is what you get when you take the category $\mathsf{Mod}_R$ of $R$-modules, and kill the projective modules. (Of course, ...

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### Generators of triangulated category and Grothendieck groups

Let $\mathcal{T}$ be a triangulated category that is generated by one object, say $A$ in the sense that the smallest triangulated subcategory containing $A$ and closed under coproducts and ...

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### Literature request: $K^b(\text{proj} A)$ Krull-Schmidt for $\text{gl dim}A = \infty$ and general results about its Grothendieck group

I'm interested in the Grothedieck group of the triangulated category $K^b(\text{proj}A)$ when $A$ is a finite dimensional algebra over a field of infinite global dimension.
For this purpose, It would ...

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### Decompose an unbounded (cochain) complex in the homotopy category

Let $\mathcal{A}$ be an abelian category, it is known that any complex $A^{\bullet}$ admits a distinguished triangle
$$B^{\bullet}\rightarrow A^{\bullet}\rightarrow C^{\bullet}\rightarrow B^{\bullet}[...

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### Mapping cone is a functor

It is a well-known general fact that in a triangulated category, the cone $Z$ of a morphism $X \longrightarrow Y$ (that means there exists a distinguished triangle $X \longrightarrow Y \longrightarrow ...

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### Cone of morphism induced by Serre duality

For a smooth projective variety $X$, Serre duality gives an exact autoequivalence on the derived category :
$$
S_X : D^\flat(X) \to D^\flat(X), \hspace{3em} S_X(-) = - \otimes \omega_X[\dim X]
$$
...

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### Autoequivalence group from semiorthogonal decomposition

Suppose we have a semiorthogonal decomposition $\mathcal{D} = \langle \mathcal{A}, \mathcal{B} \rangle$, and suppose we know fully the autoequivalence groups $\mathrm{Aut}(\mathcal{A})$ and $\mathrm{...

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### Can the Picard-graded homotopy of a nonzero object be nilpotent?

Let $\mathcal C$ be a symmetric monoidal stable category such that the thick subcategory generated by the unit is all of $\mathcal C$ -- in particular, every object is dualizable (I'm particularly ...

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### Did anybody study split homotopy cartesian squares in triangulated categories?

Let us call a commutative square
$$ \require{AMScd}
\begin{CD}
A @>{g'}>> B \\
@V{f'}VV @VV{f}V \\
C @>>{g}> D
\end{CD}
$$
in a triangulated category split homotopy ...

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### Rotation axiom for the triangulation on a derivator

I'm having some trouble following an argument in Moritz Groth's paper on Derivators, pointed derivators and stable derivators. More precisely, I'm currently stuck on the rotation axiom of the ...

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### Monoidal triangulated categories

I have a monoidal (not symmetric) triangulated category $(A,\otimes, 1)$ with unit 1.
Define $C$ the localizing subcategory of $A$ generated by the unit 1.
is $(C, \otimes, 1) $ a symmetric monoidal ...

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### Is there an elementary reason that this colocalisation map of complexes is a quasi-isomorphism?

A fact about triangulated categories is that (exact) localisation functors and so-called colocalisation functors come in pairs, making an exact localisation triangle. I've tried to come up with less ...

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### Grothendieck group of triangulated categories

Let $A$ be a full triangulated subcategory of $B$, $u:A\rightarrow B$ the corresponding embedding. Let $f:B\rightarrow A$ be a triangulated functor
satisfying:
$f\circ u = id$
Let $b \in B $, if $f(b)...

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### Finitely generated module, which is a virtually small complex, embeds into a module of finite projective dimension?

Let $R$ be a commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R) $ which is a ...

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### Classification of 2-periodic triangulated categories

Let $T$ be an algebraic triangulated (k-linear over a field, Hom-finite, idempotent-complete) category. Call $T$ 2-periodic if $\Omega^2(X) \cong X$ for all $X \in T$.
Question 1: Is there a ...

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### When is an object-preserving autoequivalence isomorphic to the identity?

Consider a triangulated category $\mathcal{T}$ and an exact autoequivalence $\Phi:\mathcal{T}\rightarrow \mathcal{T}$ such that $\Phi(F)=F$ for any object $F$ in $\mathcal{T}$, when could we say that $...

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### What is the Balmer spectrum of the p-complete stable homotopy category?

When doing computations with spectra, we first reduce to working at a prime p by using the arithmetic fracture theorem: (the homotopy groups of) a spectrum of finite type can be recovered from its ...

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### How to construct $X \oplus \Sigma X$ from $X \oplus \Sigma X \oplus \Sigma X \oplus \Sigma^2 X$ without splitting an idempotent?

Let $Z$ be an object in a stable (or triangulated/whatever) category $\mathcal C$. I believe it follows from Thomason's theorem (see The classification of triangulated subcategories) that the ...

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### Is the intersection of two compactly generated localizing subcategories still compactly generated?

Suppose we have a compactly generated triangulated category $\mathcal{T}$ such that the subcategory of compact objects $\mathcal{T}^c$ is essentially small. Let us take $\mathcal{A}, \mathcal{B}$ two ...

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### Is the Balmer spectrum of the derived category of the Balmer spectrum of finite spectra the Balmer spectrum of finite spectra?

$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\SH{SH}\DeclareMathOperator\Mod{Mod}\newcommand{\fin}{\mathrm{fin}}\newcommand{\cc}{\mathrm{c}}$I'm mainly interested in the dynamics of the Balmer ...

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### triangulated hull and thick hull of $\mathcal{O}_X,\dots,\mathcal{O}_X(n)$

Here I am dealing with the difference of triangulated hull and thick hull. Let $\mathcal{D}$ be a triangulated category and $\mathcal{E}\subset\mathcal{D}$ be a collection of objects. The triangulated ...

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### Voevodsky's motives and Deligne's systems of realizations

$\newcommand{\gm}{\mathrm{gm}}$Let $\mathbf{DM}_{\gm}(\mathbb{Q},\mathbb{Z})$ be Voevodsky's category of geometric motives over $\mathbb{Q}$ with coefficients in $\mathbb{Z}$ (e.g. as on p.124 of ...

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### Find a Morita equivalent finite cell DG category

I am trying to understand the following statement:
Suppose that $\mathcal{E}$ is a pre-triangulated proper DG category with a full exceptional collection. Then $\mathcal{E}$ is Morita equivalent to a ...

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### Cogenerators in a triangulated category

Let $\mathscr{T}$ be a triangulated category and $[1]$ be the shift functor in $\mathscr{T}$ and $X\in \mathscr{T}$ be a cogenerator i.e. if $Hom(Y,X[i])=0,i\in\mathbb{Z}$, then we have $Y=0$. My ...

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### Understanding Balmer spectra

$\DeclareMathOperator\Spec{Spec}\newcommand{\perf}{\mathrm{perf}}\DeclareMathOperator\SHC{SHC}$I have just finished reading the paper "The spectrum of prime ideals in tensor triangulated ...

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### Verdier localisation

$\newcommand{\Perf}{\operatorname{Perf}}$This is a toy example that I want to understand, I will be grateful for any help. Given a ring $R$ and $A=\Perf(R)$ the category of perfect complexes over $R$ ....

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### Yoneda Ext theorem and extensions

Consider the category of chain complexes over a ring $R$.
We can show that $\text{Ext}^1(M, N)$ classifies extensions using the triangulated category structure: the homotopy kernel of a map $N \...

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### Inducing an equivalence of $G$-equivariant categories

Suppose we have an equivalence of triangulated categories $\Phi : \mathcal{A} \to \mathcal{B}$. Let $G$ be a finite group. Are there any methods/conditions for specifying when one has an induced ...

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### On the not so clear relationship between torsion theories and localization for a newcomer

Given an hereditary torsion theory $(\mathcal{T}, \mathcal{F})$ on an abelian category $\mathcal{A}$, how we can relate this to a localization (i.e Ore localization). This is mentioned with not so ...

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### When semi-simple subcategories "extend" to hearts of t-structures?

Let $A$ be a semi-simple abelian subcategory of a triangulated category $C$ that "generates" $A$ (that is, $C$ equals its own smallest triangulated subcategory that is closed under direct ...

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### Any news about equivalences of periodic triangulated or $\infty$-categories?

There is a very old question (October 2009) Equivalence of derived categories which is not Fourier-Mukai which has been bumped by improving links to the literature in one of the answers and attracted ...

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### When is a thick subcategory the preimage of a weak Serre class under a homological functor?

Let $\pi : \mathcal T \to \mathcal A$ be a homological functor from a stable / triangulated category to an abelian category, and let $\mathcal C \subseteq \mathcal A$ be a weak Serre subcategory. Let $...

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### Minus sign in rotated triangles in Triangulated categories

Let $T$ be a triangulated category and
$$ X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} X[1]$$
an exact triangle (or distinguished triangle). TR 2 implies that then the two rotated triangles
$$...

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### Terminology: are there any names for "quotients" of cellular towers in stable categories?

A cellular tower in SH or in a "more general stable homotopy category" is a chain of morphisms $\dots X^{(n)}\stackrel{g^n}{\to} X^{(n+1)}\to \dots$ along with some more data and conditions; ...

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### On the definition and an example of silting/tilting subcategories in a triangulated categories according to a paper by Aihara and Iyama

In the paper "Silting mutation in triangulated categories" by Aihara and Iyama, I stumbled upon this nice definition( Definition 2.1) of a tilting/silting subcategory of a triangulated ...

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### construction of $K_0$-group and Karoubian completion

Let $A$ be a ring. The $K_0$ group of $A$ can be defined in most
old fashioned way as the Grothendieck group of the set of isomorphism classes
of its finitely generated projective $R$ modules, ...

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### How to prove a lemma of Rouquier on the dimension of triangulated categories?

In the paper of Rouquier on the dimension of triangulated categories (found here) lemma 3.5 says:
Lemma Let $\mathcal{T}$ be a triangulated category and let $\mathcal{T}_1$ and $\mathcal{T}_2$ be ...